
Using [1]} and [1]}, we can further express \(I_1\)  as
\(&{I_1} = {\left( {\frac{1}{{2\pi i}}} \right)^{\rm {2}}}{\left( {\frac{{{P_k}D_{{\rm {u}}k}^{ - {\alpha _k}}}}{{{m_{fk}}\left( {{\Lambda _k} + \gamma {\sigma ^2}} \right)}}} \right)^{{m_{fk}} + {m_{sk}}}}\frac{{{{\left( {{\sigma ^2}} \right)}^{{N_{Ik}} + {m_{fk}}}}}}{{\Gamma \!\left( {{m_{fk}} + {m_{sk}}} \right)}}\\&\times \!\int _{{{\cal L}_1}} {\int _{{{\cal L}_2}} {\frac{{\Gamma \!\left( {{m_{sk}} \!-\! {s_1}\!- \!{N_{Ik}}} \right)\Gamma \!\left( { - {s_1}} \right)\Gamma \! \left( {{m_{fk}} \!+\! {m_{sk}} \!+ \!{s_2}} \right)}}{{\Gamma \!\left( {{m_{sk}} \!+\! {s_2}} \right){\Gamma ^{ - 1}}\left( { - {s_2}} \right){\Gamma ^{ - 1}}\left( {{s_1} \!+ \!{N_{Ik}} \!+ \!{s_2}} \right)}}} }\\&\times \!{\left( {\frac{{{\sigma ^2}}}{{{P_{Ik}}{\eta _k}}}} \right)^{{s_1}}}{\left( {\frac{{ - {\Lambda _k}}}{{{\Lambda _k} + \gamma {\sigma ^2}}}} \right)^{{s_2}}}{\rm {d}}{s_2}{\rm {d}}{s_1}.\) 
